Preheat the oven to 350°F. In a large skillet or heavy braising pan, heat the oil and butter over medium-high heat. Brown the chicken in batches, removing it from the pan with a slotted spoon and setting it aside in a dish. Add the onion to the pan and cook it, scraping up any browned bits, for 4 to 5 minutes, or until softened. Add the garlic and cook for another minute, then add the walnuts and the stock. Return the chicken to the pan, cover, and place in the oven for 30 minutes. Stir in the pomegranate molasses, brown sugar and balsamic vinegar, stirring to coat the chicken. Bake the chicken, uncovered, for another hour, until the sauce is thickened and the chicken is cooked through. Season with salt and pepper. If the sauce needs to be thickened, remove the chicken pieces, cover them with foil, and reduce the sauce on the stovetop, skimming off any excess fat if necessary. Return the chicken to the pot, stir in the fresh pomegranate seeds and serve immediately with rice, noodles or mashed spuds. Serves 4.